# Bra-Ket Notation

Used in Quantum Mechanics.

## Notation

$$|\psi\rangle$$ - ket vector $$\in\mathcal{E}$$.

$$\langle\psi|$$ - bra vector $$\in\mathcal{E}^*$$ dual space.

$$\langle\psi|\psi\rangle$$ - bra-ket.

## Projections

### Position

$$\psi(\vec{r},t)=\langle\vec{r},t|\psi\rangle$$.

### Momentum

$$\psi(\vec{p},t)=\langle\vec{p},t|\psi\rangle$$.

### Examples

$$\langle\varphi|\psi\rangle = (\varphi,\psi)$$ is the notation for an inner product.

For position/ momentum space representation:

$$\langle\varphi|\psi\rangle = \int\varphi^*(\vec{r},t)\psi(\vec{r},t)d^3\vec{r} = \int\varphi_P^*(\vec{p},t)\psi_P(\vec{p},t)d^3\vec{p}$$.

## Properties

1. $$(|\psi\rangle)^*=\langle\psi|$$
2. $$(a|\psi\rangle)^*=a\langle\psi|$$
3. $$|a\psi\rangle=a|\psi\rangle$$
4. $$\langle a\psi|=a^*\langle\psi|$$
5. $$\langle\varphi|\psi = \overline{\langle\psi|\varphi\rangle}$$ since the complex conjugate transposes the swapped bra-kets
6. $$\langle\psi|\psi\rangle$$ is real, positive
7. Normalization of Wavefunction
8. Schwarz Inequality
9. Triangle Inequality
10. Orthogonality of States
11. Orthonormality of States
12. $$|\varphi\rangle|\psi\rangle$$ is allowed (and is shorthand for a tensor product of the spaces) only if $$|\varphi\rangle$$ and $$|\psi\rangle$$ are from different spaces (e.g. Positon vector and spin vector)
13. $$\langle\varphi|\psi\rangle$$ is a projection, or checks the overlap, of $$\psi$$ on $$\varphi$$

### Examples of Bra-Ket Algebra

• $$|\psi\rangle=3i|\varphi_1\rangle-7i|\varphi_2\rangle$$
• $$|\chi\rangle=-|\varphi_1\rangle+2i|\varphi_2\rangle$$
• $$\langle\psi|=-3i\langle\varphi_1|+7i\langle\varphi_2|$$
• $$|\psi+\chi\rangle=(-3i-1)|\varphi_1\rangle-5i|\varphi_2\rangle$$.

Created: 2023-06-25 Sun 02:30

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